Direct and Inverse Problems for the Heat Equation with a Dynamic type Boundary Condition

This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition

An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions

This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method

К ЧИСЛЕННОМУ РЕШЕНИЮ ОБРАТНЫХ ЗАДАЧ ДЛЯ ЛИНЕЙНОГО ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ

В статье рассматриваются обратные задачи для линейного параболического уравнения с неизвестными коэффициентами в правой части. К данным задачам, в частности, приводятся краевые задачи, с нелокальными условиями. Отдельно рассмотрены случаи, когда идентифицируемые коэффициенты зависят либо только от временной переменной, либо только от пространственной координаты. Предлагается методика численного решения задач с использованием метода прямых. Приводятся результаты численных экспериментов, проведенных на тестовых задача

A Finite Integration Method for A Time-Dependent Heat Source Identification of Inverse Problem

We investigate an inverse problem of reconstructing a timewise-dependent source for the heat equation. The solution of this problem is uniquely solvable, yet unstable. The inverse source problem two unknowns is reformulated to be a new form of forward problem one unknown. Furthermore, we propose that the finite integration method combined with the backward finite difference method can be used to solve the reformulated heat equation. The Tikhonov regularization method is employed to stabilize the noisy data. The proposed algorithm is not only easy to use but also can give an accurate and stable solution. Numerical result is presented and discussed.

Boundary Element Method for Solving Inverse Heat Source Problems

In this thesis, the boundary element method (BEM) is applied for solving inverse source problems for the heat equation. Through the employment of the Green’s formula and fundamental solution, the BEM naturally reduces the dimensionality of the problem by one although domain integrals are still present due to the initial condition and the heat source. We mainly consider the identification of time-dependent source for heat equation with several types of conditions such as non-local, non-classical, periodic, fixed point, time-average and integral which are considered as boundary or overdetermination conditions. Moreover, the more challenging cases of finding the space- and time-dependent heat source functions for additive and multiplicative cases are also considered. Under the above additional conditions a unique solution is known to exist, however, the inverse problems are still ill-posed since small errors in the input measurements result in large errors in the output heat source solution. Then some type of regularisation method is required to stabilise the solution. We utilise regularisation methods such as the Tikhonov regularisation with order zero, one, two, or the truncated singular value decomposition (TSVD) together with various choices of the regularisation parameter. The numerical results obtained from several benchmark test examples are presented in order to verify the efficiency of adopted computational methodology. The retrieved numerical solutions are compared with their analytical solutions, if available, or with the corresponding direct numerical solution, otherwise. Accurate and stable numerical solutions have been obtained throughout for all the inverse heat source problems considered

Simultaneous determination of time and space-dependent coefficients in a parabolic equation

This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the chosen regularization parameters

Inverse Problems of Determining Sources of the Fractional Partial Differential Equations

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary condition

Determination of an Unknown Coefficient in the Third Order Pseudoparabolic Equation with Non-Self-Adjoint Boundary Conditions

The solvability of the inverse boundary problem with an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved